# When using the half-angle formulas for trigonometric functions of alpha/2, I determine the sign based on the quadrant in which alpha lies.Determine whether the statement makes sense or does not make sense, and explain your reasoning.

Question
Trigonometric Functions
When using the half-angle formulas for trigonometric functions of $$\displaystyle\frac{\alpha}{{2}}$$, I determine the sign based on the quadrant in which $$\displaystyle\alpha$$ lies.Determine whether the statement makes sense or does not make sense, and explain your reasoning.

2020-12-17
For example:
The trigonometric function is given as, $$\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}.$$
$$\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}={\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}$$
Here, $$\displaystyle\alpha={225}^{\circ}.$$
Apply the half angle formula for the given trigonometric function, $$\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}$$ in which the angle $$\displaystyle{112.5}^{\circ}$$ lies in the quadrant (II) where sine and cosecant is positive. We have to determine the value of $$\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}$$ which is negative in second quadrant.
Therefore in half angle formula , negative sign has to be put.
$$\displaystyle{\cos{{\left(\frac{\alpha}{{2}}\right)}}}=-\sqrt{{\frac{{{1}+{\cos{\alpha}}}}{{2}}}}$$
The value of $$\displaystyle{\cos{{\left({112.5}^{\circ}\right)}}}={\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}{i}{s},$$
$$\displaystyle{\cos{{\left(\frac{{225}^{\circ}}{{2}}\right)}}}=-\sqrt{{\frac{{{1}+{\cos{{225}}}^{\circ}}}{{2}}}}$$
$$\displaystyle=\sqrt{{\frac{{{1}+{\left(-\frac{\sqrt{{2}}}{{2}}\right)}}}{{2}}}}$$
$$\displaystyle=-\sqrt{{{\left(\frac{{{2}-\sqrt{{2}}}}{{2}}\right)}}}$$

### Relevant Questions

When using the half-angle formulas for trigonometric functions of $$alpha/2$$, I determine the sign based on the quadrant in which $$alpha$$ lies.Determine whether the statement makes sense or does not make sense, and explain your reasoning.
Determine whether the statement "I’ve noticed that for sine, cosine, and tangent, the trig function for the sum of two angles is not equal to that trig function of the first angle plus that trig function of the second angle", makes sense or does not make sense, and explain your reasoning.

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
$$H_0:\mu_1=\mu_2.\mu_1>\mu_2$$
$$H_0:\mu_1<\mu_2.\mu_1=\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1<\mu_2$$
$$H_0:\mu_1=\mu_2.\mu_1\neq\mu_2$$
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The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
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(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
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At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver. (g) Find a 99% confidence interval for
$$\mu_1 - \mu_2$$.
lower limit
upper limit
(h) Explain the meaning of the confidence interval in the context of the problem.
Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver.
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Whether the statement “I noticed that depending on the values for Aand B assuming that they are not both zero, the graph of $$Ax^2 + By^2 = C$$ can represent any of the conic sections other than a parabola” “makes sense” or “does not make sense”, And explain your reasoning.